Rigorous Computations for Infinite Dimensional Nonlinear Problems

随着我们开发出越来越强大的计算工具，科学家们有责任利用新开发的计算技术来保持研究进展的步伐，有效地引导我们达到必须掌握高性能计算才能进步的地步。作为一个社会，我们不断承担越来越多的雄心勃勃的任务，从全基因组测序，到整个海洋的数值模拟和天气预报，再到纳米流体的分子水平模拟等等。然而，随着进步的步伐加快，有时我们需要退后一步，提出一个问题，我们的计算到底有多可靠？

As we develop evermore powerful computational tools, so it is incumbent upon scientists to exploit newly-developed computational techniques to maintain the pace of research progress, effectively leading us to the point where high-performance computing must be mastered to progress. As a society we continuously undertake more and more ambitious tasks, from whole genome sequencing, to whole-ocean numerical simulation and weather prediction to molecular-level simulation of nanofluids and onwards. However, while the pace of progress increases, sometimes we need to take a step back and pose the question, just how reliable are our computations? The goal of this proposal is to develop rigorous computational tools to prove, in a constructive mathematical way, the existence of qualitative phenomena in infinite dimensional nonlinear problems such as partial differential equations, variational equations and delay differential equations, which are successful widely used mathematical models that describe, explain and predict phenomena in areas as broad as physics, chemistry, biology and economics. More precisely, we are interested in answering each of the following questions. Can we mathematically demonstrate the reliability of the solutions computed using the Navier-Stokes equations or a closely related fluid dynamical model? Can we undertake the same program of research for the difficult, nonlinear molecular dynamics models thrown up by cell biological problems? More generally, can we develop algorithms to construct rigorously and automatically invariant sets of a given infinite dimensional nonlinear dynamical system? Finally, can we use computational methods to prove long-standing open problems in mathematical fields as broad as the calculus of variations and delay differential equations?